Question

use the product rule to find the derivative of (-7x^4 + 5x^5)(7e^x - 9). use e^x for e^x. you do not need to expand out your answer. question help: video message instructor submit question

Explanation:

Step1: Recall product - rule

The product rule states that if $y = u\cdot v$, then $y^\prime=u^\prime v + uv^\prime$. Let $u=-7x^{4}+5x^{5}$ and $v = 7e^{x}-9$.

Step2: Find $u^\prime$

Differentiate $u=-7x^{4}+5x^{5}$ with respect to $x$. Using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we get $u^\prime=-28x^{3}+25x^{4}$.

Step3: Find $v^\prime$

Differentiate $v = 7e^{x}-9$ with respect to $x$. Since $\frac{d}{dx}(e^{x})=e^{x}$ and $\frac{d}{dx}(c)=0$ (where $c$ is a constant), we have $v^\prime=7e^{x}$.

Step4: Apply product - rule

$y^\prime=u^\prime v+uv^\prime=(-28x^{3}+25x^{4})(7e^{x}-9)+(-7x^{4}+5x^{5})(7e^{x})$

Answer:

$(-28x^{3}+25x^{4})(7e^{x}-9)+(-7x^{4}+5x^{5})(7e^{x})$